Does the anthropometric model influence whole-body center of mass calculations in gait?

Examining whole-body center of mass (COM) motion is one of method being used to quantify dynamic balance and energy during gait. One common method for estimating the COM position is to apply an anthropometric model to a marker set and calculate the weighted sum from known segmental COM positions. Several anthropometric models are available to perform such a calculation. However, to date there has been no study of how the anthropometric model affects whole-body COM calculations during gait. This information is pertinent to researchers because the choice of anthropometric model may influence gait research findings and currently the trend is to consistently use a single model. In this study we analyzed a single stride of gait data from 103 young adult participants. We compared the whole-body COM motion calculated from 4 different anthropometric models (Plagenhoef et al., 1983; Winter, 1990; de Leva, 1996; Pavol et al., 2002). We found that anterior-posterior motion calculations are relatively unaffected by the anthropometric model. However, medial-lateral and vertical motions are significantly affected by the use of different anthropometric models. Our findings suggest that the researcher carefully choose an anthropometric model to fit their study populations when interested in medial-lateral or vertical motions of the COM. Our data can provide researchers a priori information on the model determination depending on the particular variable and how conservative they may want to be with COM comparisons between groups.     

Adaptive control of center of mass (global) motion and its joint (local) origin in gait

Dynamic gait stability can be quantified by the relationship of the motion state (i.e. the position and velocity) between the body center of mass (COM) and its base of support (BOS). Humans learn how to adaptively control stability by regulating the absolute COM motion state (i.e. its position and velocity) and/or by controlling the BOS (through stepping) in a predictable manner, or by doing both simultaneously following an external perturbation that disrupts their regular relationship. Post repeated-slip perturbation training, for instance, older adults learned to forward shift their COM position while walking with a reduced step length, hence reduced their likelihood of slip-induced falls. How and to what extent each individual joint influences such adaptive alterations is mostly unknown. A three-dimensional individualized human kinematic model was established. Based on the human model, sensitivity analysis was used to systematically quantify the influence of each lower limb joint on the COM position relative to the BOS and the step length during gait. It was found that the leading foot had the greatest effect on regulating the COM position relative to the BOS; and both hips bear the most influence on the step length. These findings could guide cost-effective but efficient fall-reduction training paradigm among older population.     

Can sacral marker approximate center of mass during gait and slip-fall recovery among community-dwelling older adults?

Falls are prevalent in older adults. Dynamic stability of body center of mass (COM) is critical for maintaining balance. A simple yet accurate tool to evaluate COM kinematics is essential to examine the COM stability. The purpose of this study was to determine the extent to which the COM position derived from body segmental analysis can be approximated by a single (sacral) marker during unperturbed (regular walking) and perturbed (gait-slip) gait. One hundred eighty seven older adults experienced an unexpected slip after approximately 10 regular walking trials. Two trials, the slip trial and the preceding regular walking trial, monitored with a motion capture system and force plates, were included in the present study. The COM positions were calculated by using the segmental analysis method wherein, the COM of all body segments was calculated to further estimate the body COM position. These body COM positions were then compared with those of the sacral marker placed at the second sacral vertebra for both trials. Results revealed that the COM positions were highly correlated with those of the sacrum׳s over the time intervals investigated for both walking (coefficient of correlation R>0.97) and slip (R>0.90) trials. There were detectable kinematic difference between the COM and the sacral for both trials. Our results indicated that the sacral marker can be used as a simple approximation of body COM for regular walking, and to somewhat a lesser extent, upon a slip. The benefits from the simplicity appear to overweigh the limitations in accuracy.     

Kinematic analysis of over-determinate biomechanical systems

In this paper, we introduce a new general method for kinematic analysis of rigid multi body systems subject to holonomic constraints. The method extends the standard analysis of kinematically determinate rigid multi body systems to the over-determinate case. This is accomplished by introducing a constrained optimisation problem with the objective function given as a function of the set of system equations that are allowed to be violated while the remaining equations define the feasible set. We show that exact velocity and acceleration analysis can also be performed by solving linear sets of equations, originating from differentiation of the Karush-Kuhn-Tucker optimality conditions. The method is applied to the analysis of an 18 degrees-of-freedom gait model where the kinematical drivers are prescribed with data from a motion capture experiment. The results show that significant differences are obtained between applying standard kinematic analysis or minimising the least-square errors on the two fully equivalent 3D gait models with only the way the experimental data is processed being different.     

Comparative analysis of methods for estimating arm segment parameters and joint torques from inverse dynamics

A common problem in the analyses of upper limb unfettered reaching movements is the estimation of joint torques using inverse dynamics. The inaccuracy in the estimation of joint torques can be caused by the inaccuracy in the acquisition of kinematic variables, body segment parameters (BSPs), and approximation in the biomechanical models. The effect of uncertainty in the estimation of body segment parameters can be especially important in the analysis of movements with high acceleration. A sensitivity analysis was performed to assess the relevance of different sources of inaccuracy in inverse dynamics analysis of a planar arm movement. Eight regression models and one water immersion method for the estimation of BSPs were used to quantify the influence of inertial models on the calculation of joint torques during numerical analysis of unfettered forward arm reaching movements. Thirteen subjects performed 72 forward planar reaches between two targets located on the horizontal plane and aligned with the median plane. Using a planar, double link model for the arm with a floating shoulder, we calculated the normalized joint torque peak and a normalized root mean square (rms) of torque at the shoulder and elbow joints. Statistical analyses quantified the influence of different BSP models on the kinetic variable variance for given uncertainty on the estimation of joint kinematics and biomechanical modeling errors. Our analysis revealed that the choice of BSP estimation method had a particular influence on the normalized rms of joint torques. Moreover, the normalization of kinetic variables to BSPs for a comparison among subjects showed that the interaction between the BSP estimation method and the subject specific somatotype and movement kinematics was a significant source of variance in the kinetic variables. The normalized joint torque peak and the normalized root mean square of joint torque represented valuable parameters to compare the effect of BSP estimation methods on the variance in the population of kinetic variables calculated across a group of subjects with different body types. We found that the variance of the arm segment parameter estimation had more influence on the calculated joint torques than the variance of the kinematics variables. This is due to the low moments of inertia of the upper limb, especially when compared with the leg. Therefore, the results of the inverse dynamics of arm movements are influenced by the choice of BSP estimation method to a greater extent than the results of gait analysis.     

Pitch control and speed limitation during overground deceleration in lemurid primates

An animal's fitness is influenced by the ability to move safely through its environment. Recent models have shown that aspects of body geometry, for example, limb length and center of mass (COM) position, appear to set limits for pitch control in cursorial quadrupeds. Models of pitch control predict that the body shape of these and certain other primates, with short forelimbs and posteriorly positioned COM, should allow them to decelerate rapidly while minimizing the risk of pitching forward. We chose to test these models in two non-cursorial lemurs: Lemur catta, the highly terrestrial ring-tailed lemur, and Eulemur fulvus, the highly arboreal brown lemur. We modeled the effects of changes in limb length and COM position on maximum decelerative potential for both species, as well as collecting data on maximal decelerations across whole strides. In both species, maximum measured decelerations fell below the range of pitch-limited deceleration values predicted by the geometric model, with the ring-tailed lemur approaching its pitch limit more closely. Both lemurs showed decelerative potential equivalent to or higher than horses, the only comparative model currently available. These data reinforce the hypothesis that a relatively simple model of body geometry can predict aspects of maximum performance in animals. In this case, it appears that the body geometry of primates is skewed toward avoiding forward pitch in maximal decelerations.     

Comparison of three methods to estimate the center of mass during balance assessment

Evaluation of postural control is generally based on the interpretation of the center of pressure (COP) and the center of mass (COM) time series. The purpose of this study is to compare three methods to estimate the COM which are based on different biomechanical considerations. These methods are: (1) the kinematic method; (2) the zero-point-to-zero-point double integration technique (GLP) and (3) the COP low-pass filter method (LPF). The COP and COM time series have been determined using an experimental setup with a force plate and a 3D kinematic system on six healthy young adult subjects during four different 30 s standing tasks: (a) quiet standing; (b) one leg standing; (c) voluntary oscillation about the ankles and (d) voluntary oscillation about the ankles and hips. To test the difference between the COM trajectories, the root mean square (RMS) differences between each method (three comparisons) were calculated. The RMS differences between kinematic-LPF and GLP-LPF are significantly larger than kinematic-GLP. Our results show that the GLP method is comparable to the kinematic method. Both agree with the unified theory of balance during upright stance. The GLP method is attractive in the clinical perspective because it requires only a force plate to determine the COP-COM variable, which has been demonstrated to have a high reliability.     

Kinematic analysis of over-determinate biomechanical systems

In this paper, we introduce a new general method for kinematic analysis of rigid multi body systems subject to holonomic constraints. The method extends the standard analysis of kinematically determinate rigid multi body systems to the over-determinate case. This is accomplished by introducing a constrained optimisation problem with the objective function given as a function of the set of system equations that are allowed to be violated while the remaining equations define the feasible set.

We show that exact velocity and acceleration analysis can also be performed by solving linear sets of equations, originating from differentiation of the Karush–Kuhn–Tucker optimality conditions.

The method is applied to the analysis of an 18 degrees-of-freedom gait model where the kinematical drivers are prescribed with data from a motion capture experiment.

The results show that significant differences are obtained between applying standard kinematic analysis or minimising the least-square errors on the two fully equivalent 3D gait models with only the way the experimental data is processed being different.     

Estimating plant allometric relationships in a power model with a minimum size of allocation

The allocation of resources among plant structures depends on size. For example, plants need to have a certain minimum size before they allocate resources into producing seeds. Furthermore, the allometric relationship between different plant structures and size has often been found to be adequately described by power functions. Allometric power functions have traditionally led to a bias when estimating and predicting e.g. seed production as a function of size using classical linear statistical methods. The statistical problems of using the linear models when estimating a power function with a threshold value have been solved but due to the relative complexity of the statistical solutions, the solutions are often not used in the ecological literature. Here, an intuitive and simple power model with a minimum size of allocation is investigated using a Bayesian estimation method on a simulated data set. The Bayesian estimation provided satisfactory estimates of the parameters in the model, and the model is suggested as a simple alternative when fitting allometric power functions to ecological data.     

Representation of planar motion of complex joints by means of rolling pairs. Application to neck motion

We propose to model planar movements between two human segments by means of rolling-without-slipping kinematic pairs. We compute the path traced by the instantaneous center of rotation (ICR) as seen from the proximal and distal segments, thus obtaining the fixed and moving centrodes, respectively. The joint motion is then represented by the rolling-without-slipping of one centrode on the other. The resulting joint kinematic model is based on the real movement and accounts for nonfixed axes of rotation; therefore it could improve current models based on revolute pairs in those cases where joint movement implies displacement of the ICR. Previous authors have used the ICR to characterize human joint motion, but they only considered the fixed centrode. Such an approach is not adequate for reproducing motion because the fixed centrode by itself does not convey information about body position. The combination of the fixed and moving centrodes gathers the kinematic information needed to reproduce the position and velocities of moving bodies. To illustrate our method, we applied it to the flexion-extension movement of the head relative to the thorax. The model provides a good estimation of motion both for position variables (mean R(pos)=0.995) and for velocities (mean R(vel)=0.958). This approach is more realistic than other models of neck motion based on revolute pairs, such as the dual-pivot model. The geometry of the centrodes can provide some information about the nature of the movement. For instance, the ascending and descending curves of the fixed centrode suggest a sequential movement of the cervical vertebrae.     

Predicted threshold against forward and backward loss of balance for perturbed walking

The biomechanical mechanisms of loss of balance have been studied before for slip condition but have not been investigated for arbitrary perturbation profiles under non-slip conditions in sagittal plane. This study aimed to determine the thresholds of center of mass (COM) velocity and position relative to the base of support (BOS) that predict forward and backward loss of balance during walking with a range of BOS perturbations. Perturbations were modeled as sinusoidal BOS motions in the vertical or anterior-posterior direction or as sagittal rotation. The human body was modeled using a seven-link model. Forward dynamics alongside with dynamic optimization were used to find the thresholds of initial COM velocity for each initial COM position that would predict forward or backward loss of balance. The effects of perturbation frequency and amplitude on these thresholds were modeled based on the simulation data. Experimental data were collected from 15 able-bodied individuals and three individuals with disability during perturbed walking. The simulation results showed similarity with the stability region reported for slip and non-slip conditions. The feasible stability region shrank when the perturbation frequency and amplitude increased, especially for larger initial COM velocities. 89.5% (70.9%) and 82.4% (68.2%) of the measured COM position and velocity combinations during low (high) perturbations were located inside the simulated limits of the stability region, for able-bodied and disabled individuals, respectively. The simulation results demonstrated the effects of different perturbation levels on the stability region. The obtained stability region can be used for developing rehabilitative programs in interactive environments.     

The influence of muscle model complexity in musculoskeletal motion modeling

A comparative study of four different muscle models in a musculoskeletal motion problem is made. The models vary in complexity from the simple input-output model to the more complex model of Hatze [1]. These models are used to solve a minimum time kicking problem using an optimal control algorithm. The results demonstrate the strong influence of the model choice on the various predicted kinematic and kinetic parameters in the problem. The study illustrates some of the advantages and disadvantages involved in trade-offs between model complexity and practicability in musculoskeletal motion studies. The results also illustrate the importance of appropriate detailed parameter estimation studies in the mathematical modeling of the musculoskeletal system.     

Neural network estimation of balance control during locomotion

Gait patterns of the elderly are often adjusted to accommodate for reduced function in the balance control system and a general reduction in skeletal muscle strength. Recent studies have demonstrated that measures related to motion of whole body center of mass (COM) can distinguish elderly individuals with balance impairment from healthy peers. Accurate COM estimation requires a multiple-segment anthropometric model, which may restrict its broad application in assessment of dynamic instability. Although temporal-distance measures and electromyography have been used in evaluation of overall gait function and determination of gait dysfunction, no studies have examined the use of gait measurements in predicting COM motion during gait. The purpose of this study was to demonstrate the effectiveness of an artificial neural network (ANN) model in mapping gait measurements onto COM motion in the frontal plane. Data from 40 subjects of varied age and balance impairment were entered into a 3-layer feed-forward model with back-propagated error correction. Bootstrap re-sampling was used to enhance the generalization accuracy of the model, using 20 re-sampling trials. The ANN model required minimal processing time (5 epochs, with 20 hidden units) and accurately mapped COM motion (R-values up to 0.89). As training proportion and number of hidden units increased, so did model accuracy. Overall, this model appears to be effective as a mapping tool for estimating balance control during locomotion. With easily obtained gait measures as input and a simple, computationally efficient architecture, the model may prove useful in clinical scenarios where electromyography equipment exists.     

Influence of body segment parameters and modeling assumptions on the estimate of center of mass trajectory

This study sought to determine the effect of inaccuracies in body segment parameters and modeling assumptions on the estimate of antero-posterior center of mass (COM) trajectory. Four different methods, one based on segmental kinematics, and three methods based on kinetic recordings were compared via simulation. Kinematic patterns (quiet stance, ankle-related sway, hip-ankle-related sway, sit-up and sit-up-sit-down) were tested with a 2D four-link model of the body and the ground reaction force vector was obtained by inverse dynamics. Errors in the estimation of body segment parameters were simulated by applying a +/-10% variation to one or more parameters at a time. These errors propagated differently to the COM estimated location between methods, between parameters within the same method, and between tasks. The kinematics-based method was the most sensitive to body segment parameters, with special regards to segment lengths and head-arms-trunk parameters. Root mean square error between estimated and simulated COM location reached 19mm in balance-related tasks and 38.3mm in sit-up-sit-down. The kinetics-based methods were largely less sensitive to inaccuracies in body segment parameters. In particular, the technique proposed by Zatsiorsky and King (J. Biomech. 31 (1998) 161), was completely insensitive to segment parameters. On the other hand the kinetics-based methods showed an intrinsic estimation error, due to the underlying model assumptions. The methods based on the double integration of horizontal force had better outcomes with tasks challenging such assumptions, with a maximal error in COM location of 15mm in the sit-up-sit-down. The method proposed by Shimba (J. Biomech. 17 (1984) 53) showed the best trade-off between sensitivity to body segment parameters and estimation performances given the ideal test conditions.     

A qualitative comparison of some diffusion models for neural activity via stochastic ordering

A number of diffusion processes have been proposed as a continuous analog of Stein's model for the subthreshold membrane potential of a neuron. Interspike intervals are then described as the first-passage-time of the corresponding diffusion model through a suitable threshold. Various biological considerations suggest the use of more sophisticated models in lieu of the Ornstein-Uhlenbeck model. However, the advantages of the additional complexity are not always clear. Comparisons among different models generally use numerical methods in specific examples without a general sensitivity analysis on the role of the model parameters. Here, we compare the distribution of interspike intervals from different models using the method of stochastic ordering. The qualitative comparison of the role of each parameter extends the results obtained from numerical simulations. One result on neurons with high positive net excitation is that the reversal potential models considered do not greatly differ from the Ornstein-Uhlenbeck model. For neurons with increased inhibition, the models give greater differences among the interspike interval distributions. In particular, when the mean trajectories are matched, the Feller model gives shorter times than the Ornstein-Uhlenbeck model but longer times than our double reversal potential model. Received: 5 August 1999 / Accepted in revised form: 8 May 2000     

Using linkage models to explore skull kinematic diversity and functional convergence in arthrodire placoderms

Biomechanical models offer a powerful set of tools for quantifying the diversity of function across fossil taxa. A computer‐based four‐bar linkage model previously developed to describe the potential feeding kinematics of Dunkleosteus terrelli is applied here to several other arthrodire placoderm taxa from different lineages. Arthrodire placoderms are a group of basal gnathostomes showing one of the earliest diversifications of jaw structures. The linkage model allows biomechanical variation to be compared across taxa, identify trends in skull morphology among arthrodires that potentially influence function and explore the role of linkage systems in the early evolution of jaw structures. The linkage model calculates various kinematic metrics including gape angle, effective mechanical advantage, and kinematic transmission coefficients. Results indicate that the arthrodire feeding system may be more diverse and complex than previously thought. A range of potential kinematic profiles among arthrodire taxa illustrate a diversity of feeding function comparable with modern teleost fishes. Previous estimates of bite force in Dunkleosteus are revised based on new morphological data. High levels of kinematic transmission among arthrodires suggest the potential for rapid gape expansion and possible suction feeding. Morphological comparisons indicate that there were several morphological solutions for obtaining these fast kinematics, which allowed different taxa to achieve similar kinematic profiles while varying other aspects of the feeding apparatus. Mapping of key morphological components of the linkage system on a general placoderm phylogeny illustrates the potential importance of four‐bar systems to the early evolution of jaw structures. J. Morphol. 271:990–1005, 2010. © 2010 Wiley‐Liss, Inc.     

Akaike information criterion should not be a “test” of geographical prediction accuracy in ecological niche modelling

Model complexity in ecological niche modelling has been recently considered as an important issue that might affect model performance. New methodological developments have implemented the Akaike information criterion (AIC) to capture model complexity in the Maxent algorithm model. AIC is calculated based on the number of parameters and likelihoods of continuous raw outputs. ENMeval R package allows users to perform a species-specific tuning of Maxent settings running models with different combinations of regularization multiplier and feature classes and finally, all these models are compared using AIC corrected for small sample size. This approach is focused to find the “best” model parametrization and it is thought to maximize the model complexity and therefore, its predictability. We found that most niche modelling studies examined by us (68%) tend to consider AIC as a criterion of predictive accuracy in geographical distribution. In other words, AIC is used as a criterion to choose those models with the highest capacity to discriminate between presences and absences. However, the link between AIC and geographical predictive accuracy has not been tested so far. Here, we evaluated this relationship using a set of simulated (virtual) species. We created a set of nine virtual species with different ecological and geographical traits (e.g., niche position, niche breadth, range size) and generated different sets of true presences and absences data across geography. We built a set of models using Maxent algorithm with different regularization values and features schemes and calculated AIC values for each model. For each model, we obtained binary predictions using different threshold criteria and validated using independent presence and absences data. We correlated AIC values against standard validation metrics (e.g., Kappa, TSS) and the number of pixels correctly predicted as presences and absences. We did not find a correlation between AIC values and predictive accuracy from validation metrics. In general, those models with the lowest AIC values tend to generate geographical predictions with high commission and omission errors. The results were consistent across all species simulated. Finally, we suggest that AIC should not be used if users are interested in prediction more than explanation in ecological niche modelling.     

Markov chain Monte Carlo segregation and linkage analysis for oligogenic models.

A new method for segregation and linkage analysis, with pedigree data, is described. Reversible jump Markov chain Monte Carlo methods are used to implement a sampling scheme in which the Markov chain can jump between parameter subspaces corresponding to models with different numbers of quantitative-trait loci (QTL's). Joint estimation of QTL number, position, and effects is possible, avoiding the problems that can arise from misspecification of the number of QTL's in a linkage analysis. The method is illustrated by use of a data set simulated for the 9th Genetic Analysis Workshop; this data set had several oligogenic traits, generated by use of a 1,497-member pedigree. The mixing characteristics of the method appear to be good, and the method correctly recovers the simulated model from the test data set. The approach appears to have great potential both for robust linkage analysis and for the answering of more general questions regarding the genetic control of complex traits.     

Structural identifiability of systems biology models: a critical comparison of methods

Analysing the properties of a biological system through in silico experimentation requires a satisfactory mathematical representation of the system including accurate values of the model parameters. Fortunately, modern experimental techniques allow obtaining time-series data of appropriate quality which may then be used to estimate unknown parameters. However, in many cases, a subset of those parameters may not be uniquely estimated, independently of the experimental data available or the numerical techniques used for estimation. This lack of identifiability is related to the structure of the model, i.e. the system dynamics plus the observation function. Despite the interest in knowing a priori whether there is any chance of uniquely estimating all model unknown parameters, the structural identifiability analysis for general non-linear dynamic models is still an open question. There is no method amenable to every model, thus at some point we have to face the selection of one of the possibilities. This work presents a critical comparison of the currently available techniques. To this end, we perform the structural identifiability analysis of a collection of biological models. The results reveal that the generating series approach, in combination with identifiability tableaus, offers the most advantageous compromise among range of applicability, computational complexity and information provided.     

Bayesian phylogenetic model selection using reversible jump Markov chain Monte Carlo

A common problem in molecular phylogenetics is choosing a model of DNA substitution that does a good job of explaining the DNA sequence alignment without introducing superfluous parameters. A number of methods have been used to choose among a small set of candidate substitution models, such as the likelihood ratio test, the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), and Bayes factors. Current implementations of any of these criteria suffer from the limitation that only a small set of models are examined, or that the test does not allow easy comparison of non-nested models. In this article, we expand the pool of candidate substitution models to include all possible time-reversible models. This set includes seven models that have already been described. We show how Bayes factors can be calculated for these models using reversible jump Markov chain Monte Carlo, and apply the method to 16 DNA sequence alignments. For each data set, we compare the model with the best Bayes factor to the best models chosen using AIC and BIC. We find that the best model under any of these criteria is not necessarily the most complicated one; models with an intermediate number of substitution types typically do best. Moreover, almost all of the models that are chosen as best do not constrain a transition rate to be the same as a transversion rate, suggesting that it is the transition/transversion rate bias that plays the largest role in determining which models are selected. Importantly, the reversible jump Markov chain Monte Carlo algorithm described here allows estimation of phylogeny (and other phylogenetic model parameters) to be performed while accounting for uncertainty in the model of DNA substitution.